Problem 72 - A Graphical Approach to Precalculus with Limits

Question

Work each problem. If \(f(3)=-9.7,\) identify a point on the graph of \(f\)

Step-by-Step Solution

Answer

The point on the graph is \((3, -9.7)\).

Step 1: Identify the Coordinates

To identify a point on the graph, we need to understand the relation given by the function notation. The notation \(f(3) = -9.7\) tells us that when the input for the function \(f\) is 3, the output is -9.7. This output represents the y-value corresponding to the x-value of 3.

Step 2: Write the Ordered Pair

Using the information from the function notation, we can write the ordered pair that represents the point on the graph. The coordinate pair is formed as \((x, y)\). Here, \(x\) is the input value, which is 3, and \(y\) is the output value, which is -9.7. Therefore, the point is \((3, -9.7)\).

Key Concepts

Function Notation

Function notation is a way to indicate a relationship between input and output values in a function. Typically expressed as \( f(x) \), it communicates that "\( f \)" is a function and "\( x \)" is the variable or input. The function notation format \( f(3) = -9.7 \) tells us that when we substitute 3 for \( x \) in the function \( f \), the result or output is \(-9.7\). Keep in mind:

• The letter "\( f \)" is commonly used, but any letter can represent a function, like \( g(x) \) or \( h(t) \).
• The input value, sometimes called the independent variable, can be any number within the domain of the function.
• The output, or dependent variable, shows how the function behaves at the input value specified.

By mastering function notation, you can easily extract specific values from a function's description and become more familiar with interpreting mathematical relationships.

Ordered Pairs

Ordered pairs are essential in representing points within a coordinate system, most commonly within the context of a graph. Each ordered pair is generally written as \((x, y)\), where:

• \( x \) is the first element, representing a position on the horizontal axis (commonly known as the x-axis).
• \( y \) is the second element, indicating a position on the vertical axis (known as the y-axis).

Using the function notation example \( f(3) = -9.7 \), we can interpret this as the ordered pair \((3, -9.7)\). Here, 3 is the x-value, meaning it is the specific input to the function, and \(-9.7\) is the y-value, representing the output. Thus, the ordered pair \((3, -9.7)\) marks a distinct point on the graph of the function. Understanding how to generate and use ordered pairs is fundamental for analyzing the relationships between variables and graphing them accurately.

Graph of a Function

The graph of a function provides a visual representation of the relationship expressed by that function. When you plot a function on a coordinate plane:

• The x-axis typically represents the input values.
• The y-axis shows the corresponding output values resulting from those inputs.
• Each point \((x, y)\) on the graph represents the output of the function at that specific input value "\( x \)".

Taking the example \( f(3) = -9.7 \), the point \((3, -9.7)\) would appear on the graph of \( f \). This point tells us that when the function is given 3 as an input, the output is \(-9.7\).A function graph can tell us:

• How the output changes as the input changes.
• The overall behavior of the function, such as growth, decrease, or any key patterns.

By interpreting the graph, one can gain insights about the function, making it easier to analyze complex mathematical situations.